Year 6 Maths reasoning (SATs): The nth term and the position-to-term rule
Part of the Year 6 Maths reasoning (SATs) algebra and sequences chapter. Chapter overview · whole chapter on one page
The nth term and the position-to-term rule
Rule
The nth term is the term in position n. For a linear sequence, the nth term is first term + (n - 1) x difference.
A formula for the nth term looks like 3n + 1. The number in front of n is the difference.
Key ideas
- nth term = first term + (n - 1) x difference.
- The number in front of n in the formula is the difference.
- To check a formula, put n = 1 and n = 2 in and compare.
Method. Start at the first term and add the difference for each step after it.
- Find the first term and the difference. The first term is 4, and the difference is 3.4 and 3
- Count the steps to the nth term. The 10th term is 9 steps from the first term.10 - 1 = 9 steps
- Work out the term. Start at 4 and add 3 for each step.4 + 27 = 31
- Write the answer. 31
Check. 4 + (10 - 1) x 3 = 31. ✓
Watch out. The 1st term has 0 steps added, so the 10th term has 9 steps.
Method. The number in front of n is the difference. Then work out what to add or subtract.
- Find the number that goes with n. The term goes up by 3 each time, so the formula has 3n.3n
- Check the first term. When n = 1, 3 x 1 + (the rest) must equal 4.3 + 1 = 4
- Write the formula. Use the number that makes the first term correct.3n + 1
- Write the answer. 3n + 1
Check. When n = 2, the term is 7. ✓
Watch out. Check your formula with n = 1 and n = 2.
Method. Check whether the difference from the first term is a multiple of the difference.
- Find the difference from the first term. 41 - 5 = 36.36
- Divide by the difference. 36 / 4 = 9, a whole number.9
- Decide. So 41 is term number 10.term 10
- Write the answer. yes
Check. The terms are 5 + 4 x (whole number). ✓
Watch out. A number is in the sequence only if it can be reached by adding the difference repeatedly.
Watch out
Watch out. The 10th term is 9 steps after the 1st term, not 10 steps.
Exam tip
Practice guide. Test your formula with n = 1. If it gives the first term, it is probably right.
Exercise 7B Practice: the nth term
Non-calculatorMethod. Start at the first term and add the difference for each step after it.
- Find the first term and the difference. The first term is 6, and the difference is 4.6 and 4
- Count the steps to the nth term. The 8th term is 7 steps from the first term.8 - 1 = 7 steps
- Work out the term. Start at 6 and add 4 for each step.6 + 28 = 34
- Write the answer. 34
Check. 6 + (8 - 1) x 4 = 34. ✓
Watch out. The 1st term has 0 steps added, so the 10th term has 9 steps.
Method. Start at the first term and add the difference for each step after it.
- Find the first term and the difference. The first term is 2, and the difference is 5.2 and 5
- Count the steps to the nth term. The 20th term is 19 steps from the first term.20 - 1 = 19 steps
- Work out the term. Start at 2 and add 5 for each step.2 + 95 = 97
- Write the answer. 97
Check. 2 + (20 - 1) x 5 = 97. ✓
Watch out. The 1st term has 0 steps added, so the 10th term has 9 steps.
Method. Start at the first term and add the difference for each step after it.
- Find the first term and the difference. The first term is 50, and the difference is -3.50 and -3
- Count the steps to the nth term. The 12th term is 11 steps from the first term.12 - 1 = 11 steps
- Work out the term. Start at 50 and add -3 for each step.50 - 33 = 17
- Write the answer. 17
Check. 50 + (12 - 1) x -3 = 17. ✓
Watch out. The 1st term has 0 steps added, so the 10th term has 9 steps.
Method. Start at the first term and add the difference for each step after it.
- Find the first term and the difference. The first term is 3, and the difference is 4.3 and 4
- Count the steps to the nth term. The 50th term is 49 steps from the first term.50 - 1 = 49 steps
- Work out the term. Start at 3 and add 4 for each step.3 + 196 = 199
- Write the answer. 199
Check. 3 + (50 - 1) x 4 = 199. ✓
Watch out. The 1st term has 0 steps added, so the 10th term has 9 steps.
Method. The number in front of n is the difference. Then work out what to add or subtract.
- Find the number that goes with n. The term goes up by 4 each time, so the formula has 4n.4n
- Check the first term. When n = 1, 4 x 1 + (the rest) must equal 7.4 + 3 = 7
- Write the formula. Use the number that makes the first term correct.4n + 3
- Write the answer. 4n + 3
Check. When n = 2, the term is 11. ✓
Watch out. Check your formula with n = 1 and n = 2.
Method. The number in front of n is the difference. Then work out what to add or subtract.
- Find the number that goes with n. The term goes up by 6 each time, so the formula has 6n.6n
- Check the first term. When n = 1, 6 x 1 + (the rest) must equal 1.6 + -5 = 1
- Write the formula. Use the number that makes the first term correct.6n - 5
- Write the answer. 6n - 5
Check. When n = 2, the term is 7. ✓
Watch out. Check your formula with n = 1 and n = 2.
Method. The number in front of n is the difference. Then work out what to add or subtract.
- Find the number that goes with n. The term goes up by -4 each time, so the formula has -4n.-4n
- Check the first term. When n = 1, -4 x 1 + (the rest) must equal 20.-4 + 24 = 20
- Write the formula. Use the number that makes the first term correct.-4n + 24
- Write the answer. -4n + 24
Check. When n = 2, the term is 16. ✓
Watch out. Check your formula with n = 1 and n = 2.
Method. Check whether the difference from the first term is a multiple of the difference.
- Find the difference from the first term. 57 - 3 = 54.54
- Divide by the difference. 54 / 6 = 9, a whole number.9
- Decide. So 57 is term number 10.term 10
- Write the answer. yes
Check. The terms are 3 + 6 x (whole number). ✓
Watch out. A number is in the sequence only if it can be reached by adding the difference repeatedly.
Method. Check whether the difference from the first term is a multiple of the difference.
- Find the difference from the first term. 52 - 4 = 48.48
- Check the division by the difference. 48 / 5 is not a whole number.not whole
- Decide. The terms never land on 52.no
- Write the answer. no
Check. The terms are 4 + 5 x (whole number). ✓
Watch out. A number is in the sequence only if it can be reached by adding the difference repeatedly.
Method. Check whether the difference from the first term is a multiple of the difference.
- Find the difference from the first term. 50 - 1 = 49.49
- Divide by the difference. 49 / 7 = 7, a whole number.7
- Decide. So 50 is term number 8.term 8
- Write the answer. yes
Check. The terms are 1 + 7 x (whole number). ✓
Watch out. A number is in the sequence only if it can be reached by adding the difference repeatedly.
Method. Test each choice against the question.
- Read what the question asks. Decide what the answer must do or show.
- Test the choice. When n = 1, 3 x 1 + 1 = 4, and when n = 2, 3 x 2 + 1 = 7, so it fits.3n + 1 fits
- Reject the other choices. n + 3 gives 4 and 5, not 7, and 4n gives 4 and 8.n + 3 and 4n do not fit
- Write the answer. 3n + 1
Check. 3n + 1 is the one that fits. ✓
Method. Check the difference from the first term.
- Difference from the first term. 90 - 4 = 86.86
- Divide by 5. 86 / 5 = 17.2, which is not a whole number.not whole
- Write the answer. No, because the difference is 5, and 90 - 4 = 86, which is not a multiple of 5.
Check. 86 / 5 is not whole. ✓
Method. Use the nth term rule with first term 5 and difference 3.
- Steps to the 20th term. The 20th term is 19 steps after the first term.19 steps
- Add. 5 + 19 x 3 = 5 + 57 = 62.62
- Write the answer. 62
Check. 5 + 57 = 62. ✓
Method. Count the steps from the first term.
- Steps. The 100th term is 99 steps after the first term.99 steps
- Add. 1 + 99 x 2 = 1 + 198 = 199.199
- Write the answer. Because the difference is 2, and 99 steps of 2 from 1 is 1 + 198 = 199.
Check. 1 + 99 x 2 = 199. ✓
Method. The difference is -3, so the formula starts with -3n.
- Find the number with n. The difference is -3.-3n
- Find the number to add. When n = 1, -3 + 15 = 12.+15
- Write the answer. -3n + 15
Check. -3 + 15 = 12, and -6 + 15 = 9. ✓